Metrical Diophantine approximation for continued fraction-like maps of the interval

Date of Completion

January 2010






For a given irrational x, the Gauss map, T( x) = 〈1/x〉, provides an infinite sequence of rational approximations, each with a corresponding approximation coefficient, &thetas;(x, pq ) = q2:x − pq :. The results we are after concern the long-term behavior of this and related sequences. The ergodicity of the Gauss map has been used to produce many such results in the classical case. ^ An infinite family of similar maps having continued fraction-like expansions is defined, and it is shown that these expansions converge. As in the classical case, the fact that these maps are not automorphisms means that the sequence of approximation coefficients cannot be described in terms of the orbit of x. For each of the maps under consideration, we define an extension that effectively recovers the lost information. It is shown that this is in fact the natural automorphic extension. From this we know that these maps are ergodic. The approximation coefficients can be described in terms of the natural extension, but not in terms of a generic point, so we cannot apply the Ergodic Theorem directly. The fact that the natural extension is expanding in the first coordinate and contracting in the second is used to prove that the conclusions of the Ergodic Theorem still hold, given mild assumptions. ^ We consider the space of approximating pairs. For the classical continued fractions, this space is a triangle: for any irrational x, the sum of any two consecutive approximation coefficients is less than 1. The analogous regions for the u-continued fractions are found. In some cases the shape of the region yields a better upper bound on the minimum of two consecutive approximation coefficients than that for the individual approximation coefficients. An interesting wrinkle occurs when the function mapping the domain of the natural extension onto the space of approximating pairs is not one-to-one, and the corresponding region is non-convex. Ergodic theory provides not only the region in which the pairs live, but the distribution of the pairs within this region, for almost all x. From the distribution of the pairs, the distribution of the approximation coefficients themselves follows directly. ^